Question: $\sum\limits_{n=2}^{\infty}\dfrac{\sqrt {n}}{\ln(n)}$ Does the integral test apply to the series? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: $\dfrac{\sqrt{x}}{\ln(x)}$ is continuous and positive for all $x\geq 2$. To find whether it's always decreasing for $x\geq2$, let's consider its derivative. $\dfrac{d}{dx}\left(\dfrac{\sqrt{x}}{\ln(x)}\right)=\dfrac{\ln(x)-2}{2\sqrt{x}\ln^2(x)}$ For $x> e^2$, we have $\ln(x)>2$, which means $\ln(x)-2$ is positive. So $\dfrac{d}{dx}\left(\dfrac{\sqrt{x}}{\ln(x)}\right)$ is positive for all $x > e^2$, which means $\dfrac{\sqrt{x}}{\ln(x)}$ is increasing on that interval. In conclusion, the integral test does not apply to the series.